Quantum Chicken Rice

TWO-LEVEL SYSTEMS

WAVE FUNCTIONS ARE NOT ENOUGH

Stern-Gerlach experiment Electron has an intrinsic angular momentum called spin Description of spin is outside domain of wave functions

STERN-GERLACH SETUP

ORBITAL MAGNETIC DIPOLE MOMENT

Magnetic dipole moment and orbital angular momentum

1

I =e

T=

ev

2⇡r(1)

this current produces a magnetic field that from far away looks just like the field from a bar magnet with a magnetic dipole moment

1

I =e

T=

ev

2⇡r(1)

µ = IA = I⇡r2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~L (4)

1

I =e

T=

ev

2⇡r(1)

µ = IA = I⇡r2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~L (4)

1

I =e

T=

ev

2⇡r(1)

µ = IA = I⇡r2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~L (4)

angular momentum

INTERACTION WITH EXTERNAL MAGNETIC FIELD

The potential energy of interaction (energy of orientation)

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = n~ (11)

µ =e~2m

n (12)

Translational force acting on the bar magnet:

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = n~ (11)

µ =e~2m

n (12)

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = n~ (11)

µ =e~2m

n (12)

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = n~ (11)

µ =e~2m

n (12)

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = n~ (11)

µ =e~2m

n (12)

CLASSICAL PREDICTION

Thermal atoms have randomly oriented magnetic moments

since the field has a gradient along the vertical axis, it is expected that atoms will get deflected along this direction and different atoms by different amount so that finally we should see on the screen something similar to the right panel

EXPERIMENTAL DATA

Magnetic moment is quantized In conclusion so is angular momentum

And this result is the same for various orientations of the magnet.

POSTCARD TO BOHR

HAS IT REALLY BEEN CONFIRMED...?

Back to Bohr model

angular momentum is quantized l = 1,2,3,...

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = l~ (11)

L

z

= m~ (12)

µ =e~2m

n (13)

suggest that the components are also quantized ml = -l, -l+1,...,0,1,..., l

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = l~ (11)

L

z

= m

l

~ (12)

µ

z

= � e~2m

m

l

= �µ

B

m

l

(13)

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = l~ (11)

L

z

= m

l

~ (12)

µ

z

= � e~2m

m

l

= �µ

B

m

l

(13)corresponding quantization of magnetic moment

Bohr magneton

1

I =e

T

=ev

2⇡r(1)

µ = IA = I⇡r

2 (2)

µ =evr

2=

e

2mL (3)

~µ = � e

2m~

L (4)

V = �~µ · ~B (5)

~

F = �~rV (6)

F

x

= ~µ · @~

B

@x

(7)

F

y

= ~µ · @~

B

@y

(8)

F

z

= ~µ · @~

B

@z

(9)

~⌧ = ~µ⇥ ~

B (10)

L = l~ (11)

L

z

= m

l

~ (12)

µ

z

= � e~2m

m

l

= �µ

B

m

l

(13)

µ

B

= 9.27⇥ 10�24 J/T (14)

Conclusion: there always is an odd number of possible values for a component of orbital magnetic dipole moment.

SPIN

There is a new source of angular momentum Stern-Gerlach result is not due to orbital angular momentum Let us postulate existence of another angular momentum

2

µ

z

= � e~2m

m

l

= �µ

B

m

l

(13)

µ

B

= 9.27⇥ 10�24 J/T (14)

~µ

s

= � e

2m~

S (15)

S

z

= m

s

~ = ±1

2~ (16)

z component of spin (intrinsic angular momentum)

from the experiment: only two opposite sign components equal to 1/2 hbar as we assume that the difference in ms is 1, just as for orbital angular momentum

ELECTRON IS SPIN 1/2 PARTICLE

Spin quantum number Assuming that allowed spin components are from -s to s, s = 1/2.

Spin gyromagnetic ratio Relation between spin magnetic moment and spin angular momentum

2

µ

z

= � e~2m

m

l

= �µ

B

m

l

(13)

µ

B

= 9.27⇥ 10�24 J/T (14)

~µ

s

= � e

2m~

S (15)

S

z

= m

s

~ = ±1

2~ (16)

s =1

2(17)

µ

s

= �g

s

µ

B

m

s

(18)

spin gyromagnetic ratio very close to 2, established experimentally

WHY IS THIS CALLED SPIN?

The Abraham-Lorentz model Electron is a charged rotating (spinning) sphere

Problems of this classical picture (including relativistic effects) 1. Spin velocity essentially equal to the speed of light 2. Unstable: radiates energy away, and fast...

SIMPLEST QUANTUM SYSTEM

Spin-1/2 has only two possible measurement outcomes

Other examples of two level systems Polarization of single photon A particle propagating along only two accessible paths

POLARIZATION

Describes the way electric vector behaves There are only two distinguishable polarization states

This is linearly polarized light as the electric vector oscillates in a plane. To be precise this is vertically polarized light. Fully distinguishable from horizontally polarized light.

POLARIZATION

This is again linearly polarized light but in another plane. Polarization distinguishable to this one is along orthogonal axis. Note that we can decompose this polarization in terms of horizontal and vertical.

There are many pairs of distinguishable polarizations

POLARIZATION

Circular polarization The phase between H and V is ±pi/2

Right-handed circular polarization. Left-handed circular polarization.

POLARIZER

Device that passes light of a specific polarization

LCD NUMERICAL DISPLAY

The black electrodes are transparent to light and if they are not charged the liquid crystals between them rotate polarization by 90 degrees, such that it goes through the second polarizer.

THE MALUS LAW

Governs intensity of light passing through two polarizers

IT IS HARD TO MAKE PHOTONS INTERACT

BACK TO MALUS

Every single photon either goes through or not This happens with certain probability given by... ... the general Born rule

2

µ

z

= � e~2m

m

l

= �µ

B

m

l

(13)

µ

B

= 9.27⇥ 10

�24J/T (14)

~µ

s

= � e

2m

~

S (15)

S

z

= m

s

~ = ±1

2

~ (16)

s =

1

2

(17)

µ

s

= �g

s

µ

B

m

s

(18)

~v = cos ✓ ~p+ sin ✓ ~p? (19)

COLLAPSE OF THE WAVE FUNCTION

Polarizer is a measuring device It has two outcomes: passes the photon or absorbs it

Collapse Whatever polarization of incoming photon if it passes through... ... it has to be consistent with the polarizer direction

ANOTHER EXAMPLE OF TWO-LEVEL SYSTEM

Photon that can only propagate horizontally or vertically Because we force it with mirrors and glasses (beam-splitters)

WINDOW GLASS IS A BEAM SPLITTER

You can see through and you can see your reflection Each single photon is superposed to go through and reflect

CLAUSER EXPERIMENT

Prediction of classical electromagnetism Both detectors may click even for small light intensity

light source

attenuator

50-50 beam splitter

QUANTUM COIN TOSSING

For weak light only one detector clicks In agreement with photon picture

Which detector will click in the next run? This cannot be predicted

What can be predicted? How many times a detector will click in many runs

QUANTUM INTERFERENCE

Interference of single quantum Photons, electrons,... interfere with themselves

When does it happen? When there are indistinguishable ways to reach detector

QUANTUM SEEING IN THE DARK

SUMMARY

Wave functions do not provide complete info of a system

It is convenient to describe 2-level system as 2d vectors